| L(s) = 1 | − 27i·3-s − 96.4i·5-s − 1.14e3·7-s − 729·9-s + 2.57e3i·11-s − 8.28e3i·13-s − 2.60e3·15-s + 2.26e4·17-s − 4.71e4i·19-s + 3.09e4i·21-s − 2.34e4·23-s + 6.88e4·25-s + 1.96e4i·27-s − 2.01e5i·29-s + 1.81e5·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.345i·5-s − 1.26·7-s − 0.333·9-s + 0.583i·11-s − 1.04i·13-s − 0.199·15-s + 1.11·17-s − 1.57i·19-s + 0.730i·21-s − 0.401·23-s + 0.880·25-s + 0.192i·27-s − 1.53i·29-s + 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5395384387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5395384387\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27iT \) |
| good | 5 | \( 1 + 96.4iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 1.14e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.57e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 8.28e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.26e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.71e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 2.34e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.01e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.81e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.61e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 6.35e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.17e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 9.78e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.95e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 8.00e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 4.79e4iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 3.61e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.53e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.84e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.20e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.26e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 9.01e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.76e5T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.679619419172189481944244922468, −8.622511480393834030359531423171, −7.64170534322223165631065922644, −6.75294339164304443491325775039, −5.87050436747374728675384311252, −4.78225477718286856899687863886, −3.34203681963633321252953806823, −2.49615040952665279326536645753, −0.940365306979325588257905652021, −0.13249593894147130001226001420,
1.43908167481386305075220104253, 3.12268088835395808816604323049, 3.55072008370034317208959758152, 4.93617483026655208918138191393, 6.12538752844398931870664508537, 6.74082373823828848516467127870, 8.078767386896787966980563301696, 9.024974484937495511666795413698, 10.01521470697836249184215845662, 10.36792605406808985264429238397
